A rotating electrical machine such as a power generator or electric motor includes a rotor, which rotates together with a rotating shaft, and a stator, which is so formed as to surround the rotor from the radial-direction outside. The stator includes a cylindrical stator core, which is so formed as to surround the rotor, and a stator frame, which covers the stator core from the radial direction.
Because of the electro-magnetic pull force that occurs as the rotor rotates, electro-magnetic vibrations occur at the stator. It is necessary to prevent the electro-magnetic vibrations from transmitting to a base on which the rotating electrical machine is placed. As for the stator of a typical rotating electrical machine, an elastic body or the like is placed between the stator core and the stator frame so that the stator core is supported through the elastic body, thereby keeping the electro-magnetic vibrations from transmitting to the outside of the stator frame.
For example, in a 2-pole turbine generator, the electro-magnetic pull force that acts on the rotor vibrates in a circular mode of vibration, which has node portions of vibrations in the circumferential direction. In the 2-pole turbine generator, there are four node portions. The circular mode of vibration moves in the circumferential direction at the same frequency as a rotating speed of the rotor.
Meanwhile, like the electro-magnetic pull force, the stator vibrates in a four-node circular natural mode of vibration. The vibration mode of the stator is substantially the same as the vibration mode of the electro-magnetic pull force of the rotor. Therefore, the stator is excited severely. In this case, as the vibration mode of the electro-magnetic pull force (excitation force) increases, so does the natural mode of vibration of the stator.
The circumferential-direction distribution of the excitation force mode is represented by formula (1) because there are four node portions. In this case, the angle of the circumferential direction is represented by θ, and the magnitude of the electro-magnetic pull force of the rotor by F0.F(θ)=F0 cos(2θ)  (1)
Meanwhile, for the circumferential-direction distribution of the circular natural mode of vibration of the stator core, if the number of node portions is n, formula (2) is realized. In this case, the displacement in the circumferential direction is represented by x, and the amplitude thereof by x0.x(θ)=x0·cos(n·θ)  (2)
Accordingly, mode excitation force Fn is represented by formula (3).
                                                                        F                n                            =                            ⁢                              (                                  F                  ·                  x                                )                                                                                        =                            ⁢                                                ∫                  0                                      2                    ⁢                    π                                                  ⁢                                                      F                    0                                    ⁢                                                            cos                      ⁡                                              (                                                  2                          ⁢                          θ                                                )                                                              ·                                          x                      0                                                        ⁢                                      cos                    ⁡                                          (                                              n                        ·                        θ                                            )                                                        ⁢                                      ⅆ                    θ                                                                                                                          =                            ⁢                              {                                                                                                    =                                                                              F                            0                                                    ·                                                      x                            0                                                                                                                                                              (                                                  n                          =                          2                                                )                                                                                                                                                                          =                          0                                                ⁢                                                                                                                                                                                      (                                                  n                          ≠                          2                                                )                                                                                                                                                    (        3        )            
That is, in any cases other than n=4, orthogonality is realized. Accordingly, the mode excitation force comes to zero. Therefore, it is possible to ignore any mode response other than n=4.
The four-node natural circular mode of vibration includes two vibration modes. According to one vibration mode, antinodes of vibration (portions where the amplitude is large) are formed on, for example, a vertical central axis of a rotating-shaft-direction cross-sectional surface of the stator core. According to the other mode, the antinodes of vibration are formed on an axis that is tilted at 45 degrees from the above vertical central axis.
As for the distribution of both vibration modes, the nodes of vibration (where the amplitude becomes zero) are replaced with the antinodes of vibration. That is, at locations where antinodes emerge according to one vibration mode, nodes emerge according to the other vibration mode. Moreover, the natural frequency of one vibration mode is equal to that of the other vibration mode.
In the 2-pole turbine generator, the above four-node natural circular mode of vibration is excited by the excitation force of the rotor having substantially the same four-node circular mode of vibration. In each natural mode of vibration, the positions of nodes remain unchanged. Accordingly, in each mode response, the positions of nodes remain unchanged, and the other portions carry out harmonic vibrations in terms of time so as to be proportional to the shape of modes. Such a form of vibration is called a standing wave.
However, as in the case of the electro-magnetic pull force of the above rotor, the stator has two natural circular modes of vibration; the positions of nodes and antinodes of the vibration modes emerge differently from each other. A vibration mode that occurs on a real stator, which is calculated by superimposing responses of the two vibration modes, is in the same shape as the four-node circular mode of vibration as in the case of the excitation force.
If the vibration mode rotates at an angular speed of the rotor, the magnitude of the amplitude of the stator remains a certain value across the entire circumference. That is, in the response of each natural mode of vibration, there are node portions of vibration. However, in a real response obtained by superimposing the responses of the two vibration modes, no node portion emerges.
In this case, since the amplitude is uniform across the circumferential-direction entire circumference of the stator, it is difficult to prevent the electro-magnetic vibrations that occur at the stator from transmitting to the base on which the rotating electrical machine is placed.
As for a method of suppressing vibrations of the rotating electrical machine's stator caused by the electro-magnetic pull force of the rotor, the following is known, for example, as disclosed in Japanese Patent Application Laid-Open Publications No. 2003-088008, the entire contents of which are incorporated herein by reference, by having a polygonal cross-sectional surface of the stator core, the structural symmetry of the stator breaks, resulting in a reduction in vibration.